In any V4 two quadratic invariants of the Riemann tensor, J4 and K4, say, are known which are functionally constant, i.e. the Hamiltonian derivatives of which vanish identically. Certain natural generalizations, J2n and K2n, of degree n of these are considered, J2n being defined in any Vm but K2n only in a V2n. It is shown by methods having purely local character that J2n is functionally constant in any V2n, and that K2n vanishes identically when n is odd or when n is even and the V2n is either conformally flat or admits a hypersurface-orthogonal Killing vector; whilst in a V4n it is functionally constant. These results follow upon a prior spinorial treatment of the special case n = 3 which leads to an explicit expression for a pseudo-vector density of which the scalar density 6 is the divergence. Finally, it is shown that for appropriate values of n there exist functionally constant invariants other than J2n and K2n

Let G be a locally compact Abelian group; Γ the dual group of G; CB(Γ) the algebra of continuous, bounded functions on Γ C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; M(G) the algebra of bounded Borel measures on G; M(G)^ the algebra of Fourier–Stieltjes transforms; and M(G)^− the completion of M(G)^ in the sup-norm topology on Γ. The object of this paper is to study the natural pairing between M(G)^ and M(Γ).

In 1979 at the Siegen Topology conference Peter Eccles in his lecture asked and in most cases answered the following

Question. For which values of n does an immersion of a closed n-dimensional manifold into Rn+1 exist with a single (n + 1)-tuple point?

The answer (see [3–5, 8]) implies the following:

Proposition (Eccles). No immersion of an even dimensional orientable manifold Mn into Rn+1 has a single (n + l)-tuple point.

1. Let F be a closed, connected, orientable surface of genus g ≥ 0 smoothly embedded in S4, and let π denote the fundamental group π1(S4 − F). Then H2(π) is a quotient of H2(S4 − F) ≅ H1(F) ≅ Z2g. If F is unknotted, that is, if there is an ambient isotopy taking F to the standardly embedded surface of genus g in S3 ⊂ S4, then π ≅ Z, so H2(π) = 0. More generally, if F is the connected sum of an unknotted surface and some 2-sphere S, then π ≅ π1 (S4 − S), so again H2(π) = 0. The question of whether H2(π) could ever be non-zero was raised in (5), Problem 4.29, and (10), Conjecture 4.13, and answered in (7) and (1). There, surfaces are constructed with H2(π)≅ Z/2, and hence, by forming connected sums, with H2(π) ≅ (Z/2)n for any positive integer n. In fact, (1) produces tori T in S4 with H2(π) ≅ Z/2, and hence surfaces of genus g with H2(π) ≅ (Z/2)g.

This paper uses some recent calculations of Conder and Bujalance (on classifying finite index group extensions of Fuchsian groups with abelian quotient and torsion free kernel) in order to determine the full automorphism groups of some cylic coverings of the line (including curves of Fermat and Lefschetz type). The answer is complete for cyclic covers that branch over three points.

We give a simple constructive proof of a geometric form of Hahn-Banach theorem, simplifying a proof in [5].

Topological surgery in dimension four is used to show a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

A time harmonic Rayleigh wave, propagating in an elastic half-space y ≥ 0, is incident on a certain impedance boundary condition on y = 0, x > 0. The resulting field consists of a reflected surface wave, scattered body waves, and a transmitted surface wave appropriate to the new boundary conditions. The elastic potentials are found exactly by Fourier transform and the Wiener-Hopf technique in the case of a slightly dissipative medium. The ψ potential is found to have a logarithmic singularity at (0,0), but the φ potential though singular is bounded there. Analytic forms are given for the amplitudes of the reflected and transmitted surface waves, and for the scattered field. The reflexion coefficient is found to have a simple form for small impedances. A uniqueness theorem, based on energy considerations, is proved.

Given a Morita context (R, V, W, S), there are functors W⊗() and hom (V, ) from R-mod to; S-mod and a natural transformation λ from the first to the second. This has an epi-mono factorization and the intermediate functor we denote by ()° with natural transformations and . The tensor functor is exact if and only if WR is flat, whilst the hom functor is exact if and only if RV is projective. We begin by determining conditions under which ()° is exact; this is Theorem 1.

A function f( , ) of two variables is said to be separately continuous if for each fixed y and each fixed x, the functions f(, y) and f(x,) are continuous functions of one variable. Separately continuous functions seem to be playing a larger rôle in analysis than formerly. In (1) there is an account of the theory of semigroups in which the multiplication is separately continuous, and in (9) various analytic theorems are obtained by the study of separately continuous functions f(s, t), where t ranges over a set T, and s ranges over a set of functions on T.

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ⩾ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α.

Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E 3 numbers, and almost all intervals [x,x + log3.51x] contain E 2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E 3 numbers is optimal up to the ϵ in the exponent. The theorem on E 2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.

From a commutative ring A, Lazard(8) has made a flat injective epimorphism: A → B of commutative rings, such that if A → C is another flat injective epimorphism of commutative rings, then there is one and only one ring morphism: B → C such that the diagram

commutes; and he shows too that B → C is a flat injective epimorphism. The main aim of the present paper is to make a similar object for not necessarily commutative rings: this is achieved thanks to the notion of an A-prering, intermediate between that of an A-bimodule and that of an A-ring. In passing, prerings are also used to construct a kind of non-commutative ring of fractions.

Let $S(t) = {1}/{\pi} \arg \zeta \big(\hh + it \big)$ be the argument of the Riemann zeta-function at the point 1/2 + it. For n ⩾ 1 and t > 0 define its iterates

$$\begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,\d\tau\, + \delta_n\,, \end{equation*}$$

$$\begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*}$$

A metric on a differentiable manifold induces a bilinear form on the tangent space at each point of the manifold. The set of tangent vectors orthogonal, with respect to this bilinear form, to the whole tangent space is a vector subspace of the tangent space: the metric is called non-degenerate or degenerate at the point according as the subspace is or is not empty. This paper is concerned with the geometry of manifolds having everywhere degenerate metrics.

Sandling(6) determined the dimension subgroups of the semidirect product of a normal abelian subgroup and a subgroup; namely if G = NT is the semidirect product of a normal abelian subgroup N and a subgroup T, then the mth dimension subgroup Dm(G) of G is equal to [N, (m – 1) G] · Dm (T) for all m ≧ 1, where

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansion

This function appears in the discussion of an asymptotic formula for the function

and also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

The well-known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.

Detailed energy calculations for CH4, SiH4 and GeH4 have been carried out using the Thomas-Fermi approximation together with a nuclear field which is averaged over all orientations about the central nucleus. The energy, including exchange, has been calculated using the unmodified Thomas-Fermi electron densities as variation functions, and in this way bond lengths and force constants for the three molecules have been obtained. The effect of exchange is very marked, agreement with experiment being quite reasonable when it is included in this way.

In order to test the accuracy of this procedure, a scale factor has been introduced and used as a variational parameter. In this refined approximation, calculations have been made for SiH4, thus enabling improved charge distributions to be obtained. Energy curves are also presented for this molecule, showing separately the variations of kinetic, potential and exchange energies with internuclear distance. It is shown that whilst the scale factor has an appreciable effect on the separate energy curves, the total energy is virtually unchanged.

Finally, the reason why the exchange energy is of considerable importance in the theory is briefly discussed, and some observations are made concerning the rather abnormal bonding in GeH4.

The theory of group representations is explored with regard to statistical applications. Ease of analysis depends rather critically on the amount of symmetry present and this point is examined in detail. ‘ Stationarity’ assumptions are considered for a finite number of variates, with examples of their use in experimental situations.